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Local Energy Weak Solutions for the Navier–Stokes Equations in the Half-Space

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Abstract

The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier–Stokes equations in the half-space \({\mathbb{R}^3_+}\). Such solutions are sometimes called Lemarié–Rieusset solutions in the whole space \({\mathbb{R}^3}\). The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz–Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical \({L^3(\mathbb{R}^3_+)}\) norm obtained by Barker and Seregin for solutions developing a singularity in finite time.

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References

  1. Abe K.: The Navier-Stokes equations in a space of bounded functions. Commun. Math. Phys. 338(2), 849–865 (2015)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  2. Abe K.: On estimates for the Stokes flow in a space of bounded functions. J. Differ. Equ. 261(3), 1756–1795 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Abe K., Giga Y.: Analyticity of the Stokes semigroup in spaces of bounded functions. Acta Math. 211(1), 1–46 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Abe K., Giga Y.: The \({L^\infty}\)-Stokes semigroup in exterior domains. J. Evol. Equ. 14(1), 1–28 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barker T., Seregin G.: A necessary condition of potential blowup for the Navier–Stokes system in half-space. Math. Ann. 369(3-4), 1327–1352 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Caffarelli L., Kohn R., Nirenberg L.: Partial regularity of suitable weak solutions of the Navier–Stokes equations. Commun. Pure Appl. Math. 35(6), 771–831 (1982)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Desch W., Hieber M., Prüss J.: L p-theory of the Stokes equation in a half space. J. Evol. Equ. 1(1), 115–142 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Escauriaza L., Seregin G., Šverák V.: Backward uniqueness for parabolic equations. Arch. Ration. Mech. Anal. 169(2), 147–157 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Escauriaza, L., Seregin, G., Šverák, V.: \({L_{3,\infty}}\)-solutions of Navier–Stokes equations and backward uniqueness. Uspekhi Mat. Nauk 58(2(350)), 3–44 (2003)

  10. Farwig R., Galdi G.P., Sohr H.: A new class of weak solutions of the Navier–Stokes equations with nonhomogeneous data. J. Math. Fluid Mech. 8(3), 423–444 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Galdi, G.P.: An Introduction to the mathematical theory of the Navier–Stokes equations. In: Steady-State Problems, 2nd edn. Springer Monographs in Mathematics. Springer, New York (2011)

  12. Gallay T., Slijepčević S.: Uniform boundedness and long-time asymptotics for the two-dimensional Navier–Stokes equations in an infinite cylinder. J. Math. Fluid Mech. 17(1), 23–46 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  13. Giga Y.: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier–Stokes system. J. Differ. Equ. 62(2), 186–212 (1986)

    Article  ADS  MATH  Google Scholar 

  14. Giga Y., Hsu P.-Y., Maekawa Y.: A Liouville theorem for the planar Navier–Stokes equations with the no-slip boundary condition and its application to a geometric regularity criterion. Comm. PDE 39(10), 1906–1935 (2014)

    Article  MATH  Google Scholar 

  15. Giga Y., Sohr H.: Abstract L p estimates for the Cauchy problem with applications to the Navier–Stokes equations in exterior domains. J. Funct. Anal. 102(1), 72–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guillod, J., Šverák, V.: Numerical investigations of non-uniqueness for the Navier–Stokes initial value problem in borderline spaces. ArXiv e-prints, Apr (2017)

  17. Hopf E.: article Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen. Math. Nachr. 4, 213– (1951)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jia H., Šverák V.: Minimal L 3-initial data for potential navier–stokes singularities. SIAM J. Math. Anal. 45(3), 1448–1459 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jia H., Šverák V.: Local-in-space estimates near initial time for weak solutions of the navier-stokes equations and forward self-similar solutions. Invent. Math. 196(1), 233–265 (2014)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Jia H., Šverák V.: Are the incompressible 3D Navier–Stokes equations locally ill-posed in the natural energy space?. J. Funct. Anal. 268(12), 3734–3766 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kikuchi, N., Seregin, G.: Weak solutions to the Cauchy problem for the Navier–Stokes equations satisfying the local energy inequality. In: Nonlinear Equations and Spectral Theory, Volume 220 of Amer. Math. Soc. Transl. Ser. 2, pp. 141–164. Amer. Math. Soc., Providence (2007)

  22. Kwon, H., Tsai, T.-P.: Global Navier–Stokes flows for non-decaying initial data with slowly decaying oscillation. arXiv preprint (2018) arXiv:1811.03249

  23. Lemarié–Rieusset, P.G.: Recent Developments in the Navier–Stokes problem, Volume 431 of Chapman & Hall/CRC Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton (2002)

  24. Leray J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin F.: A new proof of the Caffarelli–Kohn–Nirenberg theorem. Commun. Pure Appl. Math. 51(3), 241–257 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Maekawa, Y., Miura, H., Prange, C.: Estimates for the Navier–Stokes equations in the half-space for non localized data. preprint (2017)

  27. Maremonti, P., Shimizu, S.: Global existence of solutions to 2-D Navier–Stokes flow with non-decaying initial data in half-plane. ArXiv e-prints, Jan. (2018)

  28. Prange, C.: Infinite energy solutions to the Navier–Stokes equations in the half-space and applications. ArXiv e-prints, Mar (2018)

  29. Seregin G.: Navier–Stokes equations: almost \({L_{3,\infty}}\)-case. J. Math. Fluid Mech. 9(1), 34–43 (2007)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  30. Seregin, G.: A note on necessary conditions for blow-up of energy solutions to the Navier–Stokes equations. In: Parabolic Problems, Volume 80 of Progr. Nonlinear Differential Equations Appl., pp. 631–645. Birkhäuser/Springer Basel AG, Basel (2011)

  31. Seregin G.: A certain necessary condition of potential blow up for Navier–Stokes equations. Commun. Math. Phys. 312(3), 833–845 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Seregin G.A.: Local regularity of suitable weak solutions to the Navier–Stokes equations near the boundary. J. Math. Fluid Mech. 4(1), 1–29 (2002)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  33. Seregin, G.A.: Necessary conditions of potential blow up for Navier–Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 385 (Kraevye Zadachi Matematicheskoĭ Fiziki i Smezhnye Voprosy Teorii Funktsiĭ. 41):187–199, 236 (2010)

  34. Seregin, G.A., Shilkin, T.N.: The local regularity theory for the Navier–Stokes equations near the boundary. In: Proceedings of the St. Petersburg Mathematical Society. Vol. XV. Advances in Mathematical Analysis of Partial Differential Equations, Volume 232 of Amer. Math. Soc. Transl. Ser. 2, pp. 219–244. Amer. Math. Soc., Providence (2014)

  35. Seregin, G.A., Shilkin, T.N., Solonnikov, V.A.: Boundary partial regularity for the Navier–Stokes equations. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 310 (Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 35 [34]):158–190 228 (2004)

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Acknowledgements

The authors thank Tai-Peng Tsai for many helpful comments. The first author is partially supported by JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, ’Development of Concentrated Mathematical Center Linking to Wisdom of the Next Generation’, which is organized by Mathematical Institute of Tohoku University. The second author is partially supported by JSPS Grants 25707005 and 17K05312. The third author acknowledges financial support from the French Agence Nationale de la Recherche under Grant ANR-16-CE40-0027-01, as well as from the IDEX of the University of Bordeaux for the BOLIDE project.

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Authors and Affiliations

  1. Department of Mathematics, Kyoto University, Kyoto, Japan

    Yasunori Maekawa

  2. Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo, Japan

    Hideyuki Miura

  3. CNRS, UMR [5251], IMB, Université de Bordeaux, Bordeaux, France

    Christophe Prange

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  1. Yasunori Maekawa
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  2. Hideyuki Miura
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  3. Christophe Prange
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Correspondence to Christophe Prange.

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Communicated by H.-T. Yau

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Maekawa, Y., Miura, H. & Prange, C. Local Energy Weak Solutions for the Navier–Stokes Equations in the Half-Space. Commun. Math. Phys. 367, 517–580 (2019). https://doi.org/10.1007/s00220-019-03344-4

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